Learning Outcomes

• Understand what a (probability) distribution is:
  • the properties of a continuous distribution.
• Use Normal Distribution to understand/describe data
  • Be able to standardise a Normal;
  • Calculate probabilities based on Normal Distribution using R.
• Know that there are other continuous distributions useful in hypothesis testing.
• Distinguish between population, sample and sampling distributions;
• Distinguish between a standard deviation and standard error of the mean;
• Describe the Central Limit Theorem;
• Use R and Excel to calculate the standard error and probabilities associated with sampling distributions;

Types of data

• Numerical
  • Continuous: yield, weight
  • Discrete: weeds per m^2
• Categorical
  • Binary: 2 mutually exclusive categories
  • Ordinal: categories ranked in order
  • Nominal: qualitative data

Example

• The gestation period (in days) for American Simmental cattle is distributed with mean 284.3 and standard deviation 5.52. How often is a calf born a week early?

Wray et al. 1987

What is a distribution

• In our case we are generally referring to a distribution function
  • This is a function (or model) that describes the probability that a system will take on value or set of values {x}
• For any variable X, we describe probabilities by
  • Discrete variables: probability distribution function P(X=x)
  • Continuous variables: probability density function f(x)
  • Discrete and Continuous variables: cumulative density function F(x) = P(X≤x)

Properties of a Continuous Distribution

• For any continuous distribution
  • There is an infinite number of possible values;
  • These values may be within a fixed interval. For example, male human heights (in cm) belong to [54.6,272].

Human height

• Specific values in a continuous distribution have probability 0. For example, the likelihood of measuring a Simmental cow at exactly 450kg is zero. This is because there are potentially an infinite number of other weights that are higher or lower than 450 kg so we say that measuring exactly 450 has a very very small probability which is equivalent to zero
• The total of all the probabilities = must be 1. (Total area under the pdf)

The Normal Distribution

• The Normal Distribution is super important because it occurs everywhere! It naturally describes many natural phenomenon and is a great for modelling the sample mean.
• It is a symmetric bell-shaped variable with two parameters \mu and \sigma^2 such that:

X\sim{N(\mu,\sigma^2)}

The Standard Normal Curve

• The standard normal curve is one where the mean = 0, and variance = 1

X\sim{N(\mu=0,\sigma^2=1)}

# Create a sequence of values for the x-axis
x_values <- seq(-4, 4, by = 0.01)

# Create a data frame to hold these values
data_frame <- data.frame(x_values)

# Plot the standard normal curve
ggplot(data_frame, aes(x = x_values)) +
  stat_function(
    fun = dnorm, args = list(mean = 0, sd = 1),
    color = "blue"
  ) +
  labs(
    title = "Standard Normal Curve",
    x = "Z-Score",
    y = "Density"
  )

The General Normal Curve

• Simmental cattle gestation times…

# Define parameters
mean <- 284.3
sd <- 5.52

# Create data and plot
x_values <- seq(mean - 4 * sd, mean + 4 * sd, length.out = 1000)
df <- data.frame(x = x_values)

ggplot(df, aes(x = x_values)) +
  stat_function(
    fun = dnorm, args = list(mean = mean, sd = sd),
    color = "blue"
  ) +
  labs(
    title = "Normal Distribution Curve",
    x = "Gestation Period (days)",
    y = "Density"
  )

The General Normal Distribution

If X\sim{N(\mu,\sigma^2)}

• PDF

f(x | \mu, \sigma) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2} for x \in (-\infty,\infty)

• CDF

F(x)=P(X\le x)=\int_{-\infty}^x f(y)dy

Types of Normal Probabilitites

• There are 3 type of probabilities that we are interested in:
  • Tail probabilities (lower and upper) = Cumulative probabilities
  • Interval probabilities;
  • Inverse probabilities.

Normal distribution in R

Normal Operating Ranges: Interval Probabilities

In environmental monitoring and animal health, we often need to know the probability of measurements falling within expected ranges:

Examples: - Stream temperature in optimal range: P(21\le T\le 23) - Normal gestation period: P(280\le X\le 285)

# Calculate probabilities for both examples
p_temp_normal <- diff(pnorm(c(21, 23), temp_mean, temp_sd))
p_gest_normal <- diff(pnorm(c(280, 285), 284.3, 5.52))

# Create temperature plot
temp_plot <- ggplot(temp_df, aes(x = temperature)) +
  stat_function(
    fun = dnorm, args = list(mean = temp_mean, sd = temp_sd),
    color = "blue"
  ) +
  geom_area(
    data = subset(temp_df, temperature >= 21 & temperature <= 23),
    aes(y = dnorm(temperature, temp_mean, temp_sd)),
    fill = "green", alpha = 0.3
  ) +
  geom_vline(xintercept = c(21, 23), linetype = "dashed", color = "darkgreen") +
  labs(
    title = "Stream Temperature",
    subtitle = sprintf("Optimal Range: %.1f%% between 21-23°C", p_temp_normal * 100),
    x = "Temperature (°C)",
    y = "Density"
  )

# Create gestation plot
gest_plot <- ggplot(gest_df, aes(x = days)) +
  stat_function(
    fun = dnorm, args = list(mean = 284.3, sd = 5.52),
    color = "blue"
  ) +
  geom_area(
    data = subset(gest_df, days >= 280 & days <= 285),
    aes(y = dnorm(days, 284.3, 5.52)),
    fill = "green", alpha = 0.3
  ) +
  geom_vline(xintercept = c(280, 285), linetype = "dashed", color = "darkgreen") +
  labs(
    title = "Gestation Period",
    subtitle = sprintf("Normal Range: %.1f%% between 280-285 days", p_gest_normal * 100),
    x = "Days",
    y = "Density"
  )

# Display plots side by side
gridExtra::grid.arrange(temp_plot, gest_plot, ncol = 2)

# Print implications
cat("Monitoring Implications:\n")

Monitoring Implications:

cat(sprintf("- %.1f%% of temperature readings should fall in optimal range\n", p_temp_normal * 100))

- 49.5% of temperature readings should fall in optimal range

cat(sprintf("- %.1f%% of births expected during normal period\n", p_gest_normal * 100))

- 33.2% of births expected during normal period

cat("\nManagement Applications:\n")

Management Applications:

cat("1. Setting monitoring frequency\n")

1. Setting monitoring frequency

cat("2. Resource allocation planning\n")

2. Resource allocation planning

cat("3. Early intervention thresholds\n")

3. Early intervention thresholds

cat("4. Performance benchmarking\n")

4. Performance benchmarking

This analysis shows how interval probabilities help: - Define normal operating conditions - Set realistic expectations - Plan resource allocation - Design monitoring programs

Finding Critical Values for Management

When designing monitoring programs, we often need to find values that capture specific probabilities:

• What temperature should trigger interventions? P(T \le x)=0.9
• When should we flag delayed births? P(X \le x)=0.95

# Calculate critical values
temp_90 <- qnorm(0.9, temp_mean, temp_sd)
gest_95 <- qnorm(0.95, 284.3, 5.52)

# Create temperature plot with critical value
temp_plot <- ggplot(temp_df, aes(x = temperature)) +
  stat_function(
    fun = dnorm, args = list(mean = temp_mean, sd = temp_sd),
    color = "blue"
  ) +
  geom_area(
    data = subset(temp_df, temperature <= temp_90),
    aes(y = dnorm(temperature, temp_mean, temp_sd)),
    fill = "purple", alpha = 0.3
  ) +
  geom_vline(xintercept = temp_90, linetype = "dashed", color = "purple") +
  annotate("text",
    x = temp_90, y = 0.2,
    label = sprintf("90th Percentile\n%.1f°C", temp_90),
    color = "purple", hjust = -0.1
  ) +
  labs(
    title = "Water Temperature Critical Value",
    subtitle = "90% of readings fall below this threshold",
    x = "Temperature (°C)",
    y = "Density"
  )

# Create gestation plot with critical value
gest_plot <- ggplot(gest_df, aes(x = days)) +
  stat_function(
    fun = dnorm, args = list(mean = 284.3, sd = 5.52),
    color = "blue"
  ) +
  geom_area(
    data = subset(gest_df, days <= gest_95),
    aes(y = dnorm(days, 284.3, 5.52)),
    fill = "purple", alpha = 0.3
  ) +
  geom_vline(xintercept = gest_95, linetype = "dashed", color = "purple") +
  annotate("text",
    x = gest_95, y = 0.05,
    label = sprintf("95th Percentile\n%.1f days", gest_95),
    color = "purple", hjust = -0.1
  ) +
  labs(
    title = "Gestation Period Critical Value",
    subtitle = "95% of births occur before this time",
    x = "Days",
    y = "Density"
  )

# Display plots side by side
gridExtra::grid.arrange(temp_plot, gest_plot, ncol = 2)

# Print management implications
cat("Management Thresholds:\n")

Management Thresholds:

cat(sprintf("- Consider interventions when temperature exceeds %.1f°C\n", temp_90))

- Consider interventions when temperature exceeds 23.9°C

cat(sprintf("- Investigate if gestation exceeds %.1f days\n", gest_95))

- Investigate if gestation exceeds 293.4 days

cat("\nApplications:\n")

Applications:

cat("1. Setting monitoring thresholds\n")

1. Setting monitoring thresholds

cat("2. Designing intervention protocols\n")

2. Designing intervention protocols

cat("3. Resource planning\n")

3. Resource planning

cat("4. Risk assessment")

4. Risk assessment

These critical values help establish evidence-based management protocols and early warning systems.

Progress Check ✓

Let’s review what we’ve learned about probability calculations:

• We can calculate different types of probabilities:
  • Lower tail: P(X \leq x) - early warning
  • Upper tail: P(X \geq x) - critical thresholds
  • Interval: P(a \leq X \leq b) - normal ranges
  • Inverse: Finding x for given probability
• These help us:
  • Assess environmental risks
  • Set monitoring thresholds
  • Make evidence-based decisions
  • Plan interventions
• Questions to consider:
  • How do these calculations change with sample data?
  • What happens to probabilities as sample size changes?

Example

• Let’s return to our example for American Simmental cattle where X~N(284.3, 5.522),
• What is the probability of a gestation time less than 275 days.

So we need to calculate the lower Tail probability such that: P(X \le 275)

# Calculate probability of early gestation
# P(X ≤ 275) where X ~ N(284.3, 5.52²)
pnorm(275, 284.3, 5.52)

[1] 0.04601526

• One would expect around 5% of gestation times would be less than 275 days/
• Question for you: Why might this be important, how can we use these results??

vxnaghiyev - stock.adobe.com

Back to the Standard Normal Curve

• Sometimes it is useful to standardise “data” as it allows us to compare samples that are drawn from populations that may have different means and standard deviations
• Luckily for us we can standardise any general normal distribution X\sim{N(\mu,\sigma^2)} to a standard normal distribution X\sim{N(0,1)}
• This was also useful as we could use a set of standard normal tables to calculate probabilities (before computers were readily available).

P(X \le x)=P\left(\frac{X-\mu}{\sigma},\frac{x-\mu}{\sigma}\right)=P\left(Z \le \frac{x-\mu}{\sigma}\right)

Standard Normal Curve

• Example: if X\sim{N(10,9)} find P(X \le 14)

P(X \le x)=P\left(\frac{X-\mu}{\sigma},\frac{x-\mu}{\sigma}\right)=P\left(Z \le \frac{x-\mu}{\sigma}\right)

P(X \le 14)=P\left(\frac{X-10}{\sqrt9},\frac{14-10}{\sqrt9}\right)=P\left(Z \le \frac{4}{3}\right)

# Calculate P(X ≤ 14) where X ~ N(10, 9)
# Method 1: Direct calculation
prob1 <- pnorm(14, 10, 3)

# Method 2: Using standardized value
prob2 <- pnorm(4 / 3, 0, 1)

# Method 3: Using standardized value (default parameters)
prob3 <- pnorm(4 / 3)

# Display results
c(direct = prob1, standardized = prob2, default = prob3)

      direct standardized      default 
   0.9087888    0.9087888    0.9087888 

Percentiles of the Standard Normal Curve

• 1 Standard deviation from the mean = 68% of the data

Percentiles of the Standard Normal Curve

• 2 Standard deviations from the mean = 95% of the data

Percentiles of the Standard Normal Curve

• 3 Standard deviations from the mean = 99.7% of the data

Not so Normal Distributions

• Student T: The Student T distribution models a symmetric bell-shaped variable with thicker tails than a Normal.
• We say the variable X \sim t_n, with n degrees of freedom.
• It has an extra parameter = n which is related to population size
• The T distribution is used for the 1 and 2 sample T-tests which are really important in the next few weeks.

Not so Normal Distributions

• The Chi-Squared distribution models a variable which can only take positive values and is skewed in distribution.
• We say the variable X \sim \chi_n^2, with n degrees of freedom.
• The Chi-Squared distribution is used for the Chi-Squared Test which you will cover in the next few weeks.

Sampling distributions

• Rye grass root growth (in mg dry weight) follows the distribution X \sim N(300, 502).
• 1. One measurement is taken: how likely is it that the dry weight exceeds 320 mg?
• 2. 10 measurements are taken: how likely is it that the sample mean exceeds 320 mg?

SERHII BLIK - stock.adobe.com

Sampling distributions

• Here, we are dealing with 2 distributions:
  • 1. Measurement: X \sim N(300,50^2)
  • 2. Sample Mean of 10 measurements: \overline X = \frac{1}{10}\Sigma_{i=1}^{10} X_i \sim ...

How does the sampling distribution occur?

• http://onlinestatbook.com/stat_sim/sampling_dist/
• We have a population X
  • We take a sample of size n and we calculate the mean \overline x_1
  • We take another sample of size n and we calculate the mean \overline x_2
  • We take another sample of size n and we calculate the mean \overline x_3 … If we sample all possibilities, then the sampling distribution of \overline X = \frac{1}{10}\Sigma_{i=1}^{10} X_i is the distribution of \{\overline x_1, \overline x_2, \overline x_3,...\}

Distribution for a sample mean

• if X\sim{N(\mu,\sigma^2)}
• then \overline X\sim{N(\mu,\frac{\sigma^2}{n})}
• Note that we call
  • \sigma the standard deviation such that sd(X)=\sigma, and
  • \sigma/\sqrt n the standard error such that sd(\overline X)=\sigma/\sqrt n
• The standard error is important for making inference on a sample populations i.e. how close your sample mean \overline x is to the population mean \mu

Example

• Rye grass root growth (in mg dry weight) follows the distribution X \sim N(300,50^2).
  • 1. One measurement is taken: how likely is it that the dry weight exceeds 320 mg?
  • 2. 10 measurements are taken: how likely is it that the sample mean exceeds 320 mg?

Example

• 1. X = Rye grass root growth \sim N(300,50^2)

P(X>320) = P\left(\frac{X-\mu}{\sigma},\frac{x-\mu}{\sigma}\right)=P\left(\frac{X-300}{50}>\frac{320-300}{50}\right) =P(Z > 0.4) =1-P(Z < 0.4) \approx 1-0.66 = 0.34

1 - pnorm(0.4)

[1] 0.3445783

Example

• 1. \overline X = Rye grass root growth \sim N(300,\frac{50^2}{10})

P(\overline X>320) = P\left(\frac{\overline X-\mu}{\frac{\sigma}{\sqrt{n}}},\frac{x-\mu}{\frac{\sigma}{\sqrt{n}}}\right)=P\left(\frac{\overline X-300}{\frac{50}{\sqrt{10}}}>\frac{320-300}{\frac{50}{\sqrt{10}}}\right) =P(Z > 1.26) =1-P(Z < 1.26) \approx 1-0.90 = 0.10

# Calculate probability of sample mean exceeding 320mg
# P(X̄ > 320) where X̄ ~ N(300, 50²/10)
# Standardized to P(Z > 1.26)
1 - pnorm(1.26)

[1] 0.1038347

CLT in Environmental Monitoring

Let’s demonstrate how the CLT works with real environmental data:

# Set random seed for reproducibility
set.seed(123)

# Define parameters
number_of_samples <- 1000
sample_sizes <- c(5, 10, 30, 50)
distributions <- list(
  "Normal" = rnorm,
  "Exponential" = rexp,
  "Chi-Squared (df = 2)" = function(n) rchisq(n, df = 2)
)

# Function to generate sample means
generate_sample_means <- function(sample_size, number_of_samples, dist_function) {
  sapply(1:number_of_samples, function(x) mean(dist_function(sample_size)))
}

# Generate sample means for all combinations
sample_means_list <- lapply(distributions, function(dist_function) {
  lapply(sample_sizes, generate_sample_means,
    number_of_samples = number_of_samples,
    dist_function = dist_function
  )
})

# Convert to data frame
sample_means_df <- do.call(rbind, lapply(names(distributions), function(dist_name) {
  do.call(rbind, lapply(1:length(sample_sizes), function(i) {
    data.frame(
      Distribution = dist_name,
      Sample_Size = sample_sizes[i],
      Sample_Mean = sample_means_list[[dist_name]][[i]]
    )
  }))
}))

Looking Ahead: CLT and Hypothesis Testing

The CLT is fundamental to statistical inference because it tells us that:

1. Sample Means are Normally Distributed
   • Even when original data isn’t normal
   • Enables use of z-tests and t-tests
   • Supports confidence interval calculations
2. Standard Error Matters
   • Measures uncertainty in sample means
   • Decreases with larger sample sizes: SE = \frac{\sigma}{\sqrt{n}}
   • Helps determine required sample sizes
3. Coming Up in Future Lectures
   • One-sample tests: Compare means to standards
   • Two-sample tests: Compare different treatments
   • ANOVA: Compare multiple groups

Example: When testing if stream temperatures exceed regulatory limits, we’ll use:

• Sample means (normally distributed thanks to CLT)
• Standard error (to assess uncertainty)
• t-tests (based on normal distribution assumptions)

Progress Check: Probability and CLT ✓

Let’s connect what we’ve learned:

1. Probability Calculations
   • Lower tail P(X \leq x) - Early warning
   • Upper tail P(X \geq x) - Critical thresholds
   • Intervals P(a \leq X \leq b) - Normal ranges
2. Sample Means (CLT)
   • Approach normal distribution
   • More reliable with larger samples
   • Enable statistical inference
3. Applications
   • Design monitoring programs
   • Set evidence-based thresholds
   • Make reliable predictions

Key Achievement: You can now connect probability theory to practical environmental monitoring decisions! 🎯

Thanks!

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